Page 60 - Textos de Matemática Vol. 39

50 CARLOS M. DA FONSECA
This survey addresses the open problem of characterizing the set of inertias {In(H)|In(Hi) = (πi,νi,δi) and rij ≤ rankHij ≤ Rij , for 1 ≤ i < j ≤ 3} ,
by a system of linear inequalities involving the orders of the blocks, the inertias of the diagonal blocks and the ranks of the off-diagonal blocks.
2. The prescribed 2 × 2 block decomposition case
In 1981, Marques de Sa´ proposed two crucial inertial problems for the characterization described in the first section. The first problem was a general- ization of some results considered some years earlier by Loewy in [17], concerned with the possible inertias for principal submatrices of UHU∗, when U runs over the unitary matrices, for a given Hermitian matrix H. His proof is based on a generalized version for principal submatrices of the so-called Cauchy interlacing inequalities.
Theorem 2.1 ([18]). Let n,m,r,π,ν,π′,ν′ be nonnegative integers such that 1 ≤ r ≤ min{m, n}. The following conditions are equivalent:
(I) There exists an n×n Hermitian matrix with inertia (π, ν, ∗) and having an r × r principal submatrix with inertia (π′, ν′, ∗).
(II) For any n × n Hermitian matrix with inertia (π, ν, ∗) there exists an unitary matrix U such that UHU∗ has an r × r principal submatrix with inertia (π′, ν′, ∗).
(III) For any n × n Hermitian matrix with inertia (π, ν, ∗) there exists an m × n matrix S, of rank r, such that In (SHS∗) = (π′, ν′, ∗).
(IV) The following inequalities hold:
π+r−n≤π′ ≤π, ν+r−n≤ν′ ≤ν,
π′+ν′≤r, π+ν≤n.
The second problem was concerned with a full description of the possible inertias of the sum of Hermitian matrices in terms of the inertia of each element of the sum.
Theorem 2.2 ([18]). Let π, ν, δ, π1, π2, ν1, ν2 be nonnegative integers. The fol- lowing conditions are equivalent:
(I) There exist two n × n Hermitian matrices, H1 and H2, such that In (Hi) = (πi, νi, δi), for i = 1, 2, and In (H1 + H2) = (π, ν, δ).
(II) There exist two n × n diagonal matrices, D1 and D2, such that In(Di) = (πi,νi,δi), for i = 1,2, and In(D1 + D2) = (π,ν,δ).
(III) The following inequalities hold:
max{π1−ν2,π2−ν1}≤π≤π1+π2, max{ν1−π2,ν2−π1}≤ν≤ν1+ν2,
π+ν+δ=πi +νi +δi =n, for i=1,2.